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Physics Today

Readers offer their own magic moments with John Bell

Robert B. Griffiths

Published Nov 30, 2015• 0 citations
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Abstract

Reinhold Bertlmann’s article “Magic moments with John Bell” was a pleasure to read. I add a note concerning some later developments that are significant for understanding the implications of Bell’s quantum foundations work.The crucial point is that under the Bell locality hypothesis for a joint measurement as given in the article,E(a,b)=∫dλρ(λ)A(a,λ)·B(b,λ)the quantity λ that represents the hidden variable(s) is a classical, not a quantum, object, and ρ(λ) is a classical probability distribution. Such expressions only make sense in the context of quantum theory when the operators of interest commute with each other, but that is not the case when this expression is used to derive a Bell inequality.Many years ago Arthur Fine pointed out that hidden variables and Bell inequalities were “imposing requirements to make well defined precisely those probability distributions for noncommuting observables whose rejection is the very essence of quantum mechanics.”11. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291 Thus the real issue is not one of locality but, instead, the proper use of probabilities in quantum theory.Fine’s point has been confirmed by the later development of a fully consistent way of introducing probabilities in quantum mechanics.22. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. That approach has shown that when quantum mechanics is interpreted using subspaces of the Hilbert space rather than classical hidden variables to represent physical properties, it is local: There are no ghostly nonlocal influences.We cannot know how Bell would have responded to those new developments, which only came to full fruition after his untimely death. The reader willing to dig into my somewhat tedious discussion33. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. of the Einstein-Podolsky-Rosen and Bohm situation will find an explicit proof that a measurement by Alice has not the slightest effect on the spin on Bob’s side. More recent is a general argument that quantum mechanics satisfies a principle of Einstein locality: What is done to system A has no effect on system B as long as there is no interaction between the two.44. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5Those proofs of quantum locality begin by taking very seriously John von Neumann’s proposal that quantum properties correspond to Hilbert subspaces, whose projectors do not, in general, commute with one another.55. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Bell, though, assumed that such properties can be represented by classical hidden variables, which always commute. Thus Bell’s theorem teaches that if one uses classical hidden variables in place of quantum properties, the result will contradict quantum theory and disagree with experiment. Indeed, while Bell started off by pointing out deficiencies in von Neumann’s argument against hidden variables, the end result of his work is an even better and much more conclusive argument, backed by experiment, for rejecting classical hidden variables.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291, Google ScholarCrossref2. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. Google Scholar3. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. Google Scholar4. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5, Google ScholarCrossref5. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Google Scholar© 2015 American Institute of Physics.

DOI
10.1063/pt.3.2996

Related papers

Readers offer their own magic moments with John Bell

Reinhold A. Bertlmann

Bertlmann replies: I am pleased at the positive response of readers to my “Magic moments with John Bell.”I enjoyed very much the contributions of Charles Clement and Kerson Huang, who reported about their own “magic moments” with Bell. Their amusing anecdotes help complete the portrait of Bell, who was an outstanding personality indeed.I found the comments of Nicholas Bykovetz very interesting. I remember that Bell strongly sympathized with Lorentzian relativity and Fitzgerald contraction. In my opinion, the ether-based Lorentzian view of relativity is just closer to the heart of the realist that Bell was. I think it was the acceptance of Lorentz’s conception on relativity rather than Einstein’s that led Bell to the correct answer that a string between two equally accelerated spaceships will break, what is known as Bell’s spaceship paradox.11. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge U. Press (2004), p. 67. In Bell’s quote “An ‘ether’ would be the cheapest solution. But the unobservability of this ether would be disturbing,” he did not mean for the “cheapest solution” to be a derogatory phrase, but rather to mean “simple.” The “unobservability of this ether” disturbed Bell, since why should the laws of physics conspire to prevent us from identifying the ether experimentally.Regarding Robert Griffiths’s arguments, I agree with some parts but strongly disagree with others. Griffiths remarks that the specific form of the expectation value used by Bell for the joint measurement of Alice and Bob in the EPR-Bohm-Bell experiment does not make sense in the context of quantum theory with noncommuting operators. That is not the point Bell wanted to make. In his formalism, the quantum states are supplemented by hidden variables, which are governed by a classical probability distribution, in order to predetermine the measurement results.The expectation value, assigned to the local hidden-variable theory, is not applied to quantum mechanics but compared with the corresponding quantum mechanical result. Its specific form is excluded via a Bell inequality, in which a certain combination of expectation values is needed. Concentrating on the expectation value alone is not enough to distinguish local hidden variable theories from quantum mechanics; for example, the predictions of quantum mechanics can also be achieved by working with a local hidden variable theory. See reference 22. R. A. Bertlmann, J. Phys. A 47, 424007 (2014). https://doi.org/10.1088/1751-8113/47/42/424007 for a more explicit discussion.Quantum mechanics as a mathematical formalism is a theory on (mathematical) Hilbert spaces, no matter whether the quantities associated with those spaces correspond to internal degrees of freedom, like spin, or external ones, like the position. In the EPR-Bohm-Bell context, the quantum formalism contains no reference to our three-dimensional space. Nevertheless, experiments are carried out in 3D space. Alice and Bob perform joint measurements of their particles at different, very remote locations. Then this nonlocal feature turns up: A measurement by Alice on the spin of her particle does have an effect— instantaneously—on Bob’s result, in contrast to what Griffiths claims in his comments. Therefore, quantum correlations are locally inexplicable.I sympathize with some of the comments by Michael Nauenberg, who collaborated with Bell long ago on “The moral aspect of quantum mechanics.”33. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics in Honor of V. F. Weisskopf, A. De-Shalit, H. Feshbach, L. Van Hove, eds., North-Holland (1966), p. 279. In particular, Nauenberg’s comment that “experiments have revealed that the nature of reality in the quantum world is different from our experience in the classical world” is, in my opinion, the lesson we have to learn from Bell inequalities.I do not appreciate so much Nauenberg’s example of the helium atom since it distracts from the issue of nonlocality. In fact, it is at macroscopic distances where the “puzzle” arises and not at atomic distances of separation.I have a confession: I am not the realist one might expect after reading Bell’s article “Bertlmann’s socks and the nature of reality”; the world in its very foundations is much more abstract than we think with our ”anschauliche” (intuitive) concepts, to borrow Werner Heisenberg’s term. My personal feeling is that Bell’s theorem, which reveals an apparent nonlocality in nature, points to a more radical conception whose onset we do not yet have.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge U. Press (2004), p. 67. Google ScholarCrossref2. R. A. Bertlmann, J. Phys. A 47, 424007 (2014). https://doi.org/10.1088/1751-8113/47/42/424007, Google ScholarCrossref3. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics in Honor of V. F. Weisskopf, A. De-Shalit, H. Feshbach, L. Van Hove, eds., North-Holland (1966), p. 279. Google Scholar© 2015 American Institute of Physics.

Physics Today•Nov 30, 2015•1 citation

Readers offer their own magic moments with John Bell

Michael Nauenberg

I read with pleasure Reinhold Bertlmann’s article about remembrances of John Bell and his contributions to physics. Bell is famous for establishing an inequality between the prediction of quantum mechanics and local hidden-variable theories, and Bertlmann wrote that “so far, all experiments looking for violations in Bell inequalities have found them, so we have to conclude, along with John, that nature contains a nonlocality in its structure.” But that conclusion is valid only in the context of Bell’s conviction that classical “realism is the proper position for a scientist.”Experiments have revealed that the nature of reality in the quantum world is different from our experience in the classical world.11. A. Aspect, Nature 446, 866 (2007). https://doi.org/10.1038/446866a Properties like the direction of an electron spin and its position and momentum remain undetermined until a measurement has been performed,22. W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science, Prometheus Books (1999), p. 44. 9781573926942 but the colors of Bertlmann’s socks are fixed after he puts them on. Albert Einstein’s famous quotation, “spooky action at a distance,”33. A. Einstein, M. Born, H. Born, The Born–Einstein Letters: Correspondence Between Albert Einstein and Max and Hedwig Born from 1916–1955, I. Born, trans., MacMillan (1971), p. 159. that appears in Bertlmann’s amusing cartoon, is also misleading, because quantum correlations for entangled electron or photon pairs also occur at atomic distance of separation—for example, in the ground state of the helium atom. What would be spooky is if those correlations were altered when the entangled pair moved apart without further interactions, but experiments have shown that this is not the case.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. A. Aspect, Nature 446, 866 (2007). https://doi.org/10.1038/446866a, Google ScholarCrossref2. W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science, Prometheus Books (1999), p. 44. 9781573926942, Google Scholar3. A. Einstein, M. Born, H. Born, The Born–Einstein Letters: Correspondence Between Albert Einstein and Max and Hedwig Born from 1916–1955, I. Born, trans., MacMillan (1971), p. 159. Google Scholar© 2015 American Institute of Physics.

Physics Today•Nov 30, 2015•3 citations

Readers offer their own magic moments with John Bell

N. Bykovetz

I greatly enjoyed reading Reinhold Bertlmann’s article “Magic moments with John Bell.” Sprinkled with levity, it discussed Bell’s life, work, and outstanding insights and achievements in the foundations of quantum mechanics (QM). In my view, the reading should also make one acutely aware that scientists, despite our expectations of them, are sometimes still prone to the same failings that beset the rest of humanity.Bertlmann outlines very nicely Bell’s theorem and inequalities and the consequences of their violation; that work led to the subsequent startling experimental results on “entangled” quantum mechanical states, which, Bertlmann writes, showed “that nature contains a nonlocality in its structure.” Nonlocality, in Bell’s conception of it, is clearly a violation of special relativity (SR), so we have a problem. The startling thing for me is that Bell actually does appear to solve the problem, at least in principle, but at the same time he seems afraid to say so and, in fact, distances himself from the proposed solution.The essence of his ambivalence is conveyed in the following paragraph, quoted by Bertlmann:It may be that we have to admit that causal influences do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An “ether” would be the cheapest solution. But the unobservability of this ether would be disturbing. So would be the impossibility of “messages” faster than light.Clearly, faster-than-light propagation is not possible in SR. But it is permissible in the approximately 100-year-old ether-based Lorentzian theory of relativity (LR), developed by Joseph Larmor, Hendrik Lorentz, Henri Poincaré, and others. Bell was explicitly cognizant of LR and discussed it in his book Speakable and Unspeakable in Quantum Mechanics (2nd edition, Cambridge University Press, 2004). It is well known that LR and SR are totally congruent in their kinematics and dynamics, for speeds equal to or less than c. All predictions are identical. However—though one cannot find this in any standard textbooks—the LR ether theory differs from SR by allowing, without contradiction, speeds greater than c.Historically, Einstein’s SR gained ascendancy, and LR was relegated to the scrap heap of history for many reasons, but I think predominantly for the following. SR was viewed as being far more elegant and simple to develop—from the perspective of theory formation—because it followed deductively from Einstein’s two simple axioms. On the other hand, LR theory was developed from a realism perspective, where waves were thought to need a medium to wave in (the luminiferous ether).Considering that Bell was clearly aware of LR and was entertaining the idea, albeit reluctantly, of “causal influences [going] faster than light”—that is, being consonant with LR—I find it difficult to understand his expressions in the quote above about cheap solutions and disturbing features. It seems to be part of human psychology that once a set of ideas or a theory is accepted as valid for a long enough time by a community of people, as has been the case for SR, it gets transformed into belief, psychologically not unlike religious belief. Once that happens, those who would question the established dogma are treated as if the questioning was ill-intentioned, if not downright sacrilegious.© 2015 American Institute of Physics.

Physics Today•Nov 30, 2015•0 citations

Bell inequalities and hidden variables over all possible paths in a quantum system

Leffler, Warren

Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.

arXiv:1207.6352•Jul 26, 2012•0 citations

Interpretations of quantum mechanics

Jean Bricmont

Bricmont replies: I find it odd, as the reviewer of a book, to be criticized because I did not discuss a theory that was not mentioned in the book. But I’ll answer Robert Griffiths point by point.In Bohmian mechanics (BM), it is simply a mathematical fact, noted by John Bell in chapter 14 of reference 1 of the Griffiths letter, that in the delayed double-slit experiment Griffith describes, particles cannot cross a symmetry plane but instead bounce back from it.11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Offering a different theory in which particles follow a different trajectory does not refute that fact.In standard quantum mechanics (QM), particles do not have trajectories; that fact was emphasized by, among others, Richard Feynman22. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. and Lev Landau and Evgeny Lifshitz.33. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2.The theory by Griffiths, based on the idea of “consistent histories,” is therefore not standard QM but is instead, like BM, an attempt to complete QM, by adding histories that consist of real events that occur independently of any measurements made on the quantum system (a measurement means an interaction with that system that may affect what would happen to it in the absence of measurements). Unfortunately, Griffiths’s attempt runs into contradictions, as shown in particular by Sheldon Goldstein in his two-part feature for Physics Today (March 1998, page 42, and April 1998, page 38).Griffiths misses the first step in Bell’s proof of nonlocality: the Einstein-Podolsky-Rosen (EPR) dilemma that perfect correlations between distant events cannot be explained unless one supposes either that some form of action at a distance occurs or that the events are predetermined by antecedent causes. The latter assumption is not a “classical” one; it is one part of the EPR dilemma. But Bell showed with his inequalities that the assumption leads to a contradiction. Hence, nonlocality follows. For more details, see, for example, references 11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. and 44. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). and Travis Norsen’s book.ReferencesSection:ChooseTop of pageReferences <<1. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Google ScholarCrossref2. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. Google Scholar3. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2. Google Scholar4. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). Google Scholar© 2019 American Institute of Physics.

Physics Today•Oct 1, 2019•2 citations

Bell on Bohm

Sheldon Goldstein

Prologue A memorial conference for John Bell, open to the public, was held at Rutgers University around 20 years ago. I gave a short talk there on Bell's views about David Bohm's “hidden variables” formulation of quantum mechanics, a version of quantum mechanics often called the de Broglie–Bohm theory or Bohmian mechanics. This theory was in fact discovered by Louis de Broglie in 1927. In 1952, it was rediscovered and developed by Bohm, who was the first to appreciate its connection to the predictions of standard quantum mechanics. I did not publish the talk. I have decided that it would be appropriate to publish a very lightly edited version of it here, in this volume devoted to the work of Bell on the foundation of quantum mechanics. One reason for doing so is that the connection between Bell and Bohm continues to be somewhat underplayed, with the strength of his advocacy of Bohmian mechanics not properly appreciated. For example, about half of the papers in Bell's collected works on the foundations of quantum mechanics deal with Bohmian mechanics. But in his fine introduction to the revised edition of this great book [1], Alain Aspect mentions this theory in only a single sentence, and parenthetically at that. For several decades after Bell proved his nonlocality theorem, based in part on Bell's inequality, it was widely claimed that Bell had shown that hidden variables – and Bohmian mechanics in particular – were impossible, that they were incompatible with the predictions of quantum mechanics. For example, the great physicist Eugene Wigner, who, unlike most of his contemporaries, was profoundly concerned with the conceptual foundations of quantum mechanics and usually wrote on the subject with great clarity and insight, has written that This [hidden variables] is an interesting idea and even though few of us were ready to accept it, it must be admitted that the truly telling argument against it was produced as late as 1965, by J. S. Bell … This appears to give a convincing argument against the hidden variables theory.

Cambridge University Press eBooks•Aug 31, 2016•0 citations
Reinhold Bertlmann’s article “Magic moments with John Bell” was a pleasure to read. I add a note concerning some later developments that are significant for understanding the implications of Bell’s quantum foundations work.The crucial point is that under the Bell locality hypothesis for a joint measurement as given in the article,E(a,b)=∫dλρ(λ)A(a,λ)·B(b,λ)the quantity λ that represents the hidden variable(s) is a classical, not a quantum, object, and ρ(λ) is a classical probability distribution. Such expressions only make sense in the context of quantum theory when the operators of interest commute with each other, but that is not the case when this expression is used to derive a Bell inequality.Many years ago Arthur Fine pointed out that hidden variables and Bell inequalities were “imposing requirements to make well defined precisely those probability distributions for noncommuting observables whose rejection is the very essence of quantum mechanics.”11. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291 Thus the real issue is not one of locality but, instead, the proper use of probabilities in quantum theory.Fine’s point has been confirmed by the later development of a fully consistent way of introducing probabilities in quantum mechanics.22. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. That approach has shown that when quantum mechanics is interpreted using subspaces of the Hilbert space rather than classical hidden variables to represent physical properties, it is local: There are no ghostly nonlocal influences.We cannot know how Bell would have responded to those new developments, which only came to full fruition after his untimely death. The reader willing to dig into my somewhat tedious discussion33. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. of the Einstein-Podolsky-Rosen and Bohm situation will find an explicit proof that a measurement by Alice has not the slightest effect on the spin on Bob’s side. More recent is a general argument that quantum mechanics satisfies a principle of Einstein locality: What is done to system A has no effect on system B as long as there is no interaction between the two.44. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5Those proofs of quantum locality begin by taking very seriously John von Neumann’s proposal that quantum properties correspond to Hilbert subspaces, whose projectors do not, in general, commute with one another.55. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Bell, though, assumed that such properties can be represented by classical hidden variables, which always commute. Thus Bell’s theorem teaches that if one uses classical hidden variables in place of quantum properties, the result will contradict quantum theory and disagree with experiment. Indeed, while Bell started off by pointing out deficiencies in von Neumann’s argument against hidden variables, the end result of his work is an even better and much more conclusive argument, backed by experiment, for rejecting classical hidden variables.REFERENCESSection:ChooseTop of pageREFERENCES <<1. A. Fine, Phys. Rev. Lett. 48, 291 (1982). https://doi.org/10.1103/PhysRevLett.48.291, Google ScholarCrossref2. R. B. Griffiths, Stanford Encyclopedia of Philosophy (2014); http://plato.stanford.edu/entries/qm-consistent-histories. Google Scholar3. R. B. Griffiths, Consistent Quantum Theory, Cambridge U. Press (2002), sec. 23.4. Google Scholar4. R. B. Griffiths, Found. Phys. 41, 705 (2011). https://doi.org/10.1007/s10701-010-9512-5, Google ScholarCrossref5. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton U. Press (1955), sec. 3.5. Google Scholar© 2015 American Institute of Physics.
Bertlmann replies: I am pleased at the positive response of readers to my “Magic moments with John Bell.”I enjoyed very much the contributions of Charles Clement and Kerson Huang, who reported about their own “magic moments” with Bell. Their amusing anecdotes help complete the portrait of Bell, who was an outstanding personality indeed.I found the comments of Nicholas Bykovetz very interesting. I remember that Bell strongly sympathized with Lorentzian relativity and Fitzgerald contraction. In my opinion, the ether-based Lorentzian view of relativity is just closer to the heart of the realist that Bell was. I think it was the acceptance of Lorentz’s conception on relativity rather than Einstein’s that led Bell to the correct answer that a string between two equally accelerated spaceships will break, what is known as Bell’s spaceship paradox.11. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge U. Press (2004), p. 67. In Bell’s quote “An ‘ether’ would be the cheapest solution. But the unobservability of this ether would be disturbing,” he did not mean for the “cheapest solution” to be a derogatory phrase, but rather to mean “simple.” The “unobservability of this ether” disturbed Bell, since why should the laws of physics conspire to prevent us from identifying the ether experimentally.Regarding Robert Griffiths’s arguments, I agree with some parts but strongly disagree with others. Griffiths remarks that the specific form of the expectation value used by Bell for the joint measurement of Alice and Bob in the EPR-Bohm-Bell experiment does not make sense in the context of quantum theory with noncommuting operators. That is not the point Bell wanted to make. In his formalism, the quantum states are supplemented by hidden variables, which are governed by a classical probability distribution, in order to predetermine the measurement results.The expectation value, assigned to the local hidden-variable theory, is not applied to quantum mechanics but compared with the corresponding quantum mechanical result. Its specific form is excluded via a Bell inequality, in which a certain combination of expectation values is needed. Concentrating on the expectation value alone is not enough to distinguish local hidden variable theories from quantum mechanics; for example, the predictions of quantum mechanics can also be achieved by working with a local hidden variable theory. See reference 22. R. A. Bertlmann, J. Phys. A 47, 424007 (2014). https://doi.org/10.1088/1751-8113/47/42/424007 for a more explicit discussion.Quantum mechanics as a mathematical formalism is a theory on (mathematical) Hilbert spaces, no matter whether the quantities associated with those spaces correspond to internal degrees of freedom, like spin, or external ones, like the position. In the EPR-Bohm-Bell context, the quantum formalism contains no reference to our three-dimensional space. Nevertheless, experiments are carried out in 3D space. Alice and Bob perform joint measurements of their particles at different, very remote locations. Then this nonlocal feature turns up: A measurement by Alice on the spin of her particle does have an effect— instantaneously—on Bob’s result, in contrast to what Griffiths claims in his comments. Therefore, quantum correlations are locally inexplicable.I sympathize with some of the comments by Michael Nauenberg, who collaborated with Bell long ago on “The moral aspect of quantum mechanics.”33. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics in Honor of V. F. Weisskopf, A. De-Shalit, H. Feshbach, L. Van Hove, eds., North-Holland (1966), p. 279. In particular, Nauenberg’s comment that “experiments have revealed that the nature of reality in the quantum world is different from our experience in the classical world” is, in my opinion, the lesson we have to learn from Bell inequalities.I do not appreciate so much Nauenberg’s example of the helium atom since it distracts from the issue of nonlocality. In fact, it is at macroscopic distances where the “puzzle” arises and not at atomic distances of separation.I have a confession: I am not the realist one might expect after reading Bell’s article “Bertlmann’s socks and the nature of reality”; the world in its very foundations is much more abstract than we think with our ”anschauliche” (intuitive) concepts, to borrow Werner Heisenberg’s term. My personal feeling is that Bell’s theorem, which reveals an apparent nonlocality in nature, points to a more radical conception whose onset we do not yet have.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge U. Press (2004), p. 67. Google ScholarCrossref2. R. A. Bertlmann, J. Phys. A 47, 424007 (2014). https://doi.org/10.1088/1751-8113/47/42/424007, Google ScholarCrossref3. J. S. Bell, M. Nauenberg, in Preludes in Theoretical Physics in Honor of V. F. Weisskopf, A. De-Shalit, H. Feshbach, L. Van Hove, eds., North-Holland (1966), p. 279. Google Scholar© 2015 American Institute of Physics.
I read with pleasure Reinhold Bertlmann’s article about remembrances of John Bell and his contributions to physics. Bell is famous for establishing an inequality between the prediction of quantum mechanics and local hidden-variable theories, and Bertlmann wrote that “so far, all experiments looking for violations in Bell inequalities have found them, so we have to conclude, along with John, that nature contains a nonlocality in its structure.” But that conclusion is valid only in the context of Bell’s conviction that classical “realism is the proper position for a scientist.”Experiments have revealed that the nature of reality in the quantum world is different from our experience in the classical world.11. A. Aspect, Nature 446, 866 (2007). https://doi.org/10.1038/446866a Properties like the direction of an electron spin and its position and momentum remain undetermined until a measurement has been performed,22. W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science, Prometheus Books (1999), p. 44. 9781573926942 but the colors of Bertlmann’s socks are fixed after he puts them on. Albert Einstein’s famous quotation, “spooky action at a distance,”33. A. Einstein, M. Born, H. Born, The Born–Einstein Letters: Correspondence Between Albert Einstein and Max and Hedwig Born from 1916–1955, I. Born, trans., MacMillan (1971), p. 159. that appears in Bertlmann’s amusing cartoon, is also misleading, because quantum correlations for entangled electron or photon pairs also occur at atomic distance of separation—for example, in the ground state of the helium atom. What would be spooky is if those correlations were altered when the entangled pair moved apart without further interactions, but experiments have shown that this is not the case.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. A. Aspect, Nature 446, 866 (2007). https://doi.org/10.1038/446866a, Google ScholarCrossref2. W. Heisenberg, Physics and Philosophy: The Revolution in Modern Science, Prometheus Books (1999), p. 44. 9781573926942, Google Scholar3. A. Einstein, M. Born, H. Born, The Born–Einstein Letters: Correspondence Between Albert Einstein and Max and Hedwig Born from 1916–1955, I. Born, trans., MacMillan (1971), p. 159. Google Scholar© 2015 American Institute of Physics.
I greatly enjoyed reading Reinhold Bertlmann’s article “Magic moments with John Bell.” Sprinkled with levity, it discussed Bell’s life, work, and outstanding insights and achievements in the foundations of quantum mechanics (QM). In my view, the reading should also make one acutely aware that scientists, despite our expectations of them, are sometimes still prone to the same failings that beset the rest of humanity.Bertlmann outlines very nicely Bell’s theorem and inequalities and the consequences of their violation; that work led to the subsequent startling experimental results on “entangled” quantum mechanical states, which, Bertlmann writes, showed “that nature contains a nonlocality in its structure.” Nonlocality, in Bell’s conception of it, is clearly a violation of special relativity (SR), so we have a problem. The startling thing for me is that Bell actually does appear to solve the problem, at least in principle, but at the same time he seems afraid to say so and, in fact, distances himself from the proposed solution.The essence of his ambivalence is conveyed in the following paragraph, quoted by Bertlmann:It may be that we have to admit that causal influences do go faster than light. The role of Lorentz invariance in the completed theory would then be very problematic. An “ether” would be the cheapest solution. But the unobservability of this ether would be disturbing. So would be the impossibility of “messages” faster than light.Clearly, faster-than-light propagation is not possible in SR. But it is permissible in the approximately 100-year-old ether-based Lorentzian theory of relativity (LR), developed by Joseph Larmor, Hendrik Lorentz, Henri Poincaré, and others. Bell was explicitly cognizant of LR and discussed it in his book Speakable and Unspeakable in Quantum Mechanics (2nd edition, Cambridge University Press, 2004). It is well known that LR and SR are totally congruent in their kinematics and dynamics, for speeds equal to or less than c. All predictions are identical. However—though one cannot find this in any standard textbooks—the LR ether theory differs from SR by allowing, without contradiction, speeds greater than c.Historically, Einstein’s SR gained ascendancy, and LR was relegated to the scrap heap of history for many reasons, but I think predominantly for the following. SR was viewed as being far more elegant and simple to develop—from the perspective of theory formation—because it followed deductively from Einstein’s two simple axioms. On the other hand, LR theory was developed from a realism perspective, where waves were thought to need a medium to wave in (the luminiferous ether).Considering that Bell was clearly aware of LR and was entertaining the idea, albeit reluctantly, of “causal influences [going] faster than light”—that is, being consonant with LR—I find it difficult to understand his expressions in the quote above about cheap solutions and disturbing features. It seems to be part of human psychology that once a set of ideas or a theory is accepted as valid for a long enough time by a community of people, as has been the case for SR, it gets transformed into belief, psychologically not unlike religious belief. Once that happens, those who would question the established dogma are treated as if the questioning was ill-intentioned, if not downright sacrilegious.© 2015 American Institute of Physics.
Bell's theorem rests on the following fundamental condition for a local system: P(a,b|alpha, beta, lambda)= P(a|alpha, lambda)P(b|beta, lambda). Here a and b are the outcomes respectively for measurements alpha on one side, and beta on the other, of an experiment involving two entangled particles traveling in opposite directions from a source. The parameter lambda (the set of "hidden variables") represents a more complete description of the joint state of the two particles. Because of lambda, the joint probability of detection is now dependent only on lambda and the local measurement setting of alpha; similarly for the other side and the setting beta. From this equation John Bell derived a simple inequality that is violated by the predictions of quantum mechanics, which is generally taken to imply that quantum mechanics is a nonlocal theory. But, by combining Richard Feynman's formulation of quantum mechanics with a model of particle interaction described by David Deutsch, we develop a system (the "space of all paths," SP) that (1) is immediately seen to replicate the predictions of quantum mechanics, (2) has a single outcome for each quantum event (unlike MWI on which it is partly based), and (3) contains the set lambda of hidden variables consisting of all possible paths from the source to the detectors on each side of the two-particle experiment. However, the set lambda is nonmeasurable, and therefore the above equation is meaningless in SP.
Bricmont replies: I find it odd, as the reviewer of a book, to be criticized because I did not discuss a theory that was not mentioned in the book. But I’ll answer Robert Griffiths point by point.In Bohmian mechanics (BM), it is simply a mathematical fact, noted by John Bell in chapter 14 of reference 1 of the Griffiths letter, that in the delayed double-slit experiment Griffith describes, particles cannot cross a symmetry plane but instead bounce back from it.11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Offering a different theory in which particles follow a different trajectory does not refute that fact.In standard quantum mechanics (QM), particles do not have trajectories; that fact was emphasized by, among others, Richard Feynman22. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. and Lev Landau and Evgeny Lifshitz.33. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2.The theory by Griffiths, based on the idea of “consistent histories,” is therefore not standard QM but is instead, like BM, an attempt to complete QM, by adding histories that consist of real events that occur independently of any measurements made on the quantum system (a measurement means an interaction with that system that may affect what would happen to it in the absence of measurements). Unfortunately, Griffiths’s attempt runs into contradictions, as shown in particular by Sheldon Goldstein in his two-part feature for Physics Today (March 1998, page 42, and April 1998, page 38).Griffiths misses the first step in Bell’s proof of nonlocality: the Einstein-Podolsky-Rosen (EPR) dilemma that perfect correlations between distant events cannot be explained unless one supposes either that some form of action at a distance occurs or that the events are predetermined by antecedent causes. The latter assumption is not a “classical” one; it is one part of the EPR dilemma. But Bell showed with his inequalities that the assumption leads to a contradiction. Hence, nonlocality follows. For more details, see, for example, references 11. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. and 44. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). and Travis Norsen’s book.ReferencesSection:ChooseTop of pageReferences <<1. J. Bricmont, S. Goldstein, D. Hemmick, J. Stat. Phys. (2019), doi: https://doi.org/10.1007/s10955-019-02361-w. Google ScholarCrossref2. R. Feynman, The Character of Physical Law, MIT Press (1967), p. 145. Google Scholar3. L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 2nd ed., J. B. Sykes, J. S. Bell, trans., Pergamon Press (1965), p. 2. Google Scholar4. R. Tumulka, “The assumptions of Bell’s proof,” International Journal of Quantum Foundations (21 December 2014). Google Scholar© 2019 American Institute of Physics.
Prologue A memorial conference for John Bell, open to the public, was held at Rutgers University around 20 years ago. I gave a short talk there on Bell's views about David Bohm's “hidden variables” formulation of quantum mechanics, a version of quantum mechanics often called the de Broglie–Bohm theory or Bohmian mechanics. This theory was in fact discovered by Louis de Broglie in 1927. In 1952, it was rediscovered and developed by Bohm, who was the first to appreciate its connection to the predictions of standard quantum mechanics. I did not publish the talk. I have decided that it would be appropriate to publish a very lightly edited version of it here, in this volume devoted to the work of Bell on the foundation of quantum mechanics. One reason for doing so is that the connection between Bell and Bohm continues to be somewhat underplayed, with the strength of his advocacy of Bohmian mechanics not properly appreciated. For example, about half of the papers in Bell's collected works on the foundations of quantum mechanics deal with Bohmian mechanics. But in his fine introduction to the revised edition of this great book [1], Alain Aspect mentions this theory in only a single sentence, and parenthetically at that. For several decades after Bell proved his nonlocality theorem, based in part on Bell's inequality, it was widely claimed that Bell had shown that hidden variables – and Bohmian mechanics in particular – were impossible, that they were incompatible with the predictions of quantum mechanics. For example, the great physicist Eugene Wigner, who, unlike most of his contemporaries, was profoundly concerned with the conceptual foundations of quantum mechanics and usually wrote on the subject with great clarity and insight, has written that This [hidden variables] is an interesting idea and even though few of us were ready to accept it, it must be admitted that the truly telling argument against it was produced as late as 1965, by J. S. Bell … This appears to give a convincing argument against the hidden variables theory.